I collected tree ring data from 4 regions across California and performed analyses on detrended growth during drought years where water availability (SPEI) was in the bottom 10th percentile. All tree ring series were spline detrended.
\[ Growth Reduction = \frac{\overline{RWI}_{non-drought} - RWI_{drought}}{\overline{RWI}_{non-drought}} \] \[ Recovery = RWI_{actual} - RWI_{pred} \]
Where \(RWI_{pred}\) is the predicted growth for a tree given the post-drought year conditions based on hierarchical Bayesian models of all growth years.
This approach differs in that here we explore the results of the full expression of the model to answer the above questions i.e. the most complex justifiable model. Each species is modeled independently. This approach is a way to avoid the model selection issues we discussed regarding the complexity of question 3.
\[ Reduction_i \sim Normal(\mu_i, \sigma_i) \]
\[ \mu_i = \alpha_{region_i} + \alpha_{treeID} + \beta_{1region_i}SPEI_i + \beta_{2}DBH_i + \beta_{3}BA_i + \beta_{4}SPEI_i\times DBH_i + \beta_{5}SPEI_i \times BA_i \]
and for recovery:
\[ Recovery_i \sim Normal(\mu_i, \sigma_i) \]
\[ \mu_i = \alpha_{region_i} + \alpha_{treeID} + \beta_{1region_i}Reduction_i + \beta_{2}DBH_i + \beta_{3}BA_i + \beta_{4}Reduction_i\times DBH_i + \beta_{5}Reduction_i \times BA_i \]
Regionally varying responses to drought intensity or severity could reflect local adaptation.
\(SPEI \times DBH\) smaller or larger trees experience the same drought intensity differently. Perhaps this is due to differences in evaporative demand, root:shoot allocation patterns, height water relationship
\(Reduction \times DBH\) smaller or larger trees recover from equally damaging droughts differently. Potentially due to access to resources, allocation again.
\(SPEI \times BA\) sparse or dense neighborhoods modify the effect of drought intensity. This could be via resource (water) competition. Although because we lack good characterization of the between neighborhood microsite differences density could reflect suitability more than competition.
\(Reduction \times BA\) sparse or dense neighborhoods modify the recovery of trees from equally damaging droughts. This could also be via resource competition although more than just water. Again, patterns may reflect uncharacterized microsite differences rather than competition.
All models converged Rhat ~1 and each model had few divergent samples. Param estimates and plots were prepared with tidybayes. Marginal effects plots were produced with add_fitted_draws() .
| waic_diff | se | |
|---|---|---|
| SPEI only | 0.00 | 0.00 |
| Species + SPEI | 1.78 | 3.06 |
| Species * SPEI | 3.86 | 3.71 |
| waic_diff | se | |
|---|---|---|
| Species var | 0.00 | 0.00 |
| Global var | 46.08 | 15.61 |
| Hypothesis | Estimate | Est.Error | CI.Lower | CI.Upper | Evid.Ratio | Post.Prob | Star |
|---|---|---|---|---|---|---|---|
| PJ - AC > 0 | 0.0678695 | 0.0095051 | 0.0524760 | 0.0835043 | Inf | 1.00000 |
|
| PL - AC > 0 | 0.0356420 | 0.0115867 | 0.0161894 | 0.0539996 | 1332.3333333 | 0.99925 |
|
| PL - PJ > 0 | -0.1274074 | 0.0467587 | -0.2027858 | -0.0502629 | 0.0025063 | 0.00250 |
| Hypothesis | Estimate | Est.Error | CI.Lower | CI.Upper | Evid.Ratio | Post.Prob | Star |
|---|---|---|---|---|---|---|---|
| PJ - AC > 0 | 0.0203995 | 0.0129695 | -0.0006292 | 0.0417334 | 16.7777778 | 0.94375 | |
| PL - AC > 0 | 0.0101130 | 0.0163853 | -0.0170997 | 0.0375752 | 2.7313433 | 0.73200 | |
| PL - PJ > 0 | -0.0102865 | 0.0164266 | -0.0374510 | 0.0168000 | 0.3633265 | 0.26650 |
| waic_diff | se | |
|---|---|---|
| Varying intercept and slope | 0.00 | 0.00 |
| Varying intercept | 5.51 | 9.90 |
| No species effect | 12.47 | 11.84 |
| waic_diff | se | |
|---|---|---|
| Species var | 0.00 | 0.0 |
| Global var | 33.67 | 20.2 |
| Hypothesis | Estimate | Est.Error | CI.Lower | CI.Upper | Evid.Ratio | Post.Prob | Star |
|---|---|---|---|---|---|---|---|
| PJ - AC > 0 | 0.0555744 | 0.0085943 | 0.0411071 | 0.0695725 | Inf | 1.00000 |
|
| PL - AC > 0 | 0.0359705 | 0.0108185 | 0.0188759 | 0.0544338 | 3.99900e+03 | 0.99975 |
|
| PL - PJ > 0 | -0.0902395 | 0.0498857 | -0.1706638 | -0.0067311 | 4.00416e-02 | 0.03850 |
| Hypothesis | Estimate | Est.Error | CI.Lower | CI.Upper | Evid.Ratio | Post.Prob | Star |
|---|---|---|---|---|---|---|---|
| PJ - AC > 0 | 0.0314717 | 0.0178364 | 0.0018100 | 0.0604578 | 25.31579 | 0.96200 |
|
| PL - AC > 0 | 0.0965939 | 0.0213785 | 0.0609767 | 0.1319395 | Inf | 1.00000 |
|
| PL - PJ > 0 | 0.0651223 | 0.0216478 | 0.0292203 | 0.1005338 | 3999.00000 | 0.99975 |
|
| Hypothesis | Estimate | Est.Error | CI.Lower | CI.Upper | Evid.Ratio | Post.Prob | Star |
|---|---|---|---|---|---|---|---|
| PJ - AC > 0 | -0.0767700 | 0.0413998 | -0.1449606 | -0.0097026 | 0.0309278 | 0.03000 | |
| PL - AC > 0 | 0.0820008 | 0.0527384 | -0.0049174 | 0.1673384 | 15.7364017 | 0.94025 | |
| PL - PJ > 0 | 0.1587708 | 0.0505887 | 0.0746240 | 0.2418538 | 570.4285714 | 0.99825 |
|
| waic_diff | se | |
|---|---|---|
| Varying intercept and slope | 0.00000 | 0.00000 |
| No regional effects | 43.30953 | 12.80732 |
| Varying intercept | 45.11601 | 12.72891 |
| waic_diff | se | |
|---|---|---|
| Varying intercept and slope | 0.00000 | 0.00000 |
| Varying intercept | 41.83552 | 12.88420 |
| No regional effects | 43.82051 | 13.91644 |
| waic_diff | se | |
|---|---|---|
| Varying intercept and slope | 0.00000 | 0.000000 |
| No regional effects | 18.41681 | 7.836080 |
| Varying intercept | 19.29546 | 7.449942 |
## # Random Effect Variances and ICC
##
## Conditioned on: ~(1 | Region)
##
## ## Variance Ratio (comparable to ICC)
## Ratio: 0.69 CI 95%: [0.48 0.81]
##
## ## Variances of Posterior Predicted Distribution
## Conditioned on fixed effects: 0.08 CI 95%: [0.06 0.15]
## Conditioned on rand. effects: 0.26 CI 95%: [0.16 0.41]
##
## ## Difference in Variances
## Difference: 0.18 CI 95%: [0.08 0.31]
## # Random Effect Variances and ICC
##
## Conditioned on: ~(1 | Region)
##
## ## Variance Ratio (comparable to ICC)
## Ratio: 0.59 CI 95%: [0.36 0.73]
##
## ## Variances of Posterior Predicted Distribution
## Conditioned on fixed effects: 0.11 CI 95%: [0.09 0.16]
## Conditioned on rand. effects: 0.27 CI 95%: [0.17 0.42]
##
## ## Difference in Variances
## Difference: 0.16 CI 95%: [0.06 0.30]
## # Random Effect Variances and ICC
##
## Conditioned on: ~(1 | Region)
##
## ## Variance Ratio (comparable to ICC)
## Ratio: 0.34 CI 95%: [-0.01 0.65]
##
## ## Variances of Posterior Predicted Distribution
## Conditioned on fixed effects: 0.11 CI 95%: [0.07 5.67]
## Conditioned on rand. effects: 0.21 CI 95%: [0.11 5.76]
##
## ## Difference in Variances
## Difference: 0.08 CI 95%: [-0.03 0.20]
| waic_diff | se | |
|---|---|---|
| Varying intercept and slope | 0.000000 | 0.000000 |
| Varying intercept | 1.851447 | 5.964911 |
| No regional effects | 12.912958 | 9.351119 |
| waic_diff | se | |
|---|---|---|
| Varying intercept and slope | 0.000000 | 0.000000 |
| Varying intercept | 7.988191 | 8.352711 |
| No regional effects | 22.441717 | 12.987328 |
| waic_diff | se | |
|---|---|---|
| Varying intercept and slope | 0.000000 | 0.000000 |
| Varying intercept | 4.836886 | 8.154183 |
| No regional effects | 36.244175 | 11.872226 |
## # Random Effect Variances and ICC
##
## Conditioned on: ~(1 | Region)
##
## ## Variance Ratio (comparable to ICC)
## Ratio: 0.05 CI 95%: [-0.09 0.19]
##
## ## Variances of Posterior Predicted Distribution
## Conditioned on fixed effects: 0.06 CI 95%: [0.05 0.09]
## Conditioned on rand. effects: 0.06 CI 95%: [0.05 0.09]
##
## ## Difference in Variances
## Difference: 0.00 CI 95%: [-0.01 0.01]
## # Random Effect Variances and ICC
##
## Conditioned on: ~(1 | Region)
##
## ## Variance Ratio (comparable to ICC)
## Ratio: 0.03 CI 95%: [-0.09 0.14]
##
## ## Variances of Posterior Predicted Distribution
## Conditioned on fixed effects: 0.10 CI 95%: [0.08 0.17]
## Conditioned on rand. effects: 0.10 CI 95%: [0.08 0.17]
##
## ## Difference in Variances
## Difference: 0.00 CI 95%: [-0.01 0.02]
## # Random Effect Variances and ICC
##
## Conditioned on: ~(1 | Region)
##
## ## Variance Ratio (comparable to ICC)
## Ratio: 0.04 CI 95%: [-0.15 0.20]
##
## ## Variances of Posterior Predicted Distribution
## Conditioned on fixed effects: 0.08 CI 95%: [0.06 0.65]
## Conditioned on rand. effects: 0.08 CI 95%: [0.06 0.64]
##
## ## Difference in Variances
## Difference: 0.00 CI 95%: [-0.02 0.02]
Keep in mind that in the scatter plots the drought intensity axis is flipped, so slope will appear to be wrong sign from the parameter estimate plots.